AUSTIN, Texas/NEW YORK, Oct 6 (Reuters) - Chalking up employment losses last month to the temporary hit of a severe hurricane season and reiterating expectations that inflation will strengthen, Federal Reserve policymakers on Friday signaled they continue to see gradual U.S. interest-rate hikes ahead.

"Even though inflation is currently somewhat below our longer-run objective, I judge that it is still appropriate to continue to remove monetary policy accommodation gradually," said New York Fed President William Dudley, whose regular meetings with Fed Chair Janet Yellen and constant contact with Wall Street banks bolster his influence among Fed policymakers.

While other policymakers largely agreed, they also said they were keeping a close eye on the data, particularly on inflation.

And one offered a strong rebuttal, saying the central bank risked a "policy mistake" if it continues raising rates despite inflation data that remains stalled.

"If we go too far in our zeal to normalize (rates) we might push inflation expectations down further and that might hinder our ability to hit our target," said St. Louis Fed President James Bullard, who called the September jobs number "startling" even given the hurricane. "The December meeting is going to be too early to make a determination on whether inflation is coming back."

Others were more on board with the December increase, though they also offered some skepticism about inflation.

Atlanta Fed President Raphael Bostic, the newest of the 12 Fed presidents, told Reuters in an interview that he continues to believe the U.S. central bank should raise interest rates again by the end of the year, though he is "not wedded" to that position and continues to track data closely.

And Robert Kaplan, chief of the Dallas Fed, told reporters that inflation is "likely building" given the low unemployment rate, which would make the case for further rate hikes. Though the number of jobs fell in September for the first time in seven years, the unemployment rate fell to 4.2 percent and hourly wages rose more than expected.

Striking a somewhat less eager tone than his colleagues though, Kaplan said, "I'm going to watch a little bit here. We have the benefit of having a little time and I plan to take it."

Last month, the Fed left rates unchanged and announced the well-telegraphed start to a gradual shrinking of its $4.5 trillion balance sheet, which was swollen by massive purchases of Treasury bonds and mortgage-backed securities in the aftermath of the 2007-2009 financial crisis and recession.

But market expectations are high that the Fed will hike rates again in December, especially after Fed Chair Janet Yellen outlined why she is fairly confident that inflation, now at 1.4 percent by the Fed's preferred measure, will rise toward the Fed's 2-percent target over the medium term. It would be imprudent, she said in late September, to wait until inflation actually reached that target to raise rates.

Investors are more skeptical of the Fed's forecasts of roughly three more hikes next year.

Three of the policymakers suggested they would be open-minded about the economic data, and especially inflation readings, for the next several months due to temporary factors weighing on prices and also the hurricanes that struck the United States over the last 40 days.

U.S. President Donald Trump recently interviewed at least three candidates who could replace Yellen when her term as Fed chief expires in February, though he is also reported to be considering reappointing her. Trump said last week that he would have a decision in the next two or three weeks.

Bostic, who started his job four months ago and has a vote next year on the central bank's rate-setting committee, said he expects the Fed, regardless of who leads it, to continue to raise rates in "a slow, steady return to more normal levels" in 2018, "absent some sign that either the economy weakens dramatically or suddenly, or if it accelerates faster than we might expect."

TAX DEBATE

Bostic said his economic forecasts do not include any changes to fiscal policy, in line with many of his colleagues.

Some Fed policymakers have begun to push back on the Trump administration's assertion that its tax cut plan would boost the economy, cautioning it could instead trigger high inflation, unsustainable debt and an eventual return to sub-par growth.

The proposed tax overhaul includes lowering the corporate tax rate to 20 percent from the current 35 percent.

"If they use it to invest, then we might see some more robust growth and then we would have to keep an eye on what happens with the level of prices," Bostic said.

"But we've had other episodes where cash windfalls have been used to buy back stocks and those sorts of things, in which case you are not getting to the same level of productive transmission of that policy."